If \(D>0\)  and \(z_{xx}>0\) , then according to Sylvester's criterion the Hessian \(Hz(x,y)\)  is positive definite and the surface has a local minimum.
Similarly, if \(D>0\)  and \(z_{xx}<0\) , then the surface has a local maximum.
If \(D<0\) , then the eigenvalues of \(Hz(x,y)\)  are of opposite sign, meaning the surface has a saddle point.
If \(D=0\) , then at least one of the eigenvalues of \(Hz(x,y)\)  is zero, yielding the test inconclusive [1]}, [2]}.
That means that there is no curvature in at least one direction, \(i.e.\)  L-shaped curve or flat surface.
Any point that is not a local minimum will be discarded.
Also Lowe discard a point if it does not meet the curvature criterion [3]}.
